Skip to main content
SpringerLink
Log in

Microstructure Effects on Effective Gas Diffusion Coefficient of Nanoporous Materials

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

In this work, we develop a numerical framework for gas diffusion in nanoporous materials including a random generation-growth algorithm for microstructure reconstruction and a multiple-relaxation-time lattice Boltzmann method for solution of diffusion equation with Knudsen effects carefully considered. The Knudsen diffusion is accurately captured by a local diffusion coefficient computed based on a corrected Bosanquet-type formula with the local pore size determined by the largest sphere method. A robust validation of the new framework is demonstrated by predicting the effective gas diffusion coefficient of microporous layer and catalyst layer in fuel cell, which shows good agreement with several recent experimental measurements. Then, a detailed investigation is made of the influence on effective gas Knudsen diffusivity by many important microstructure factors including morphology category, size effect, structure anisotropy, and layering structure effect. A widely applicable Bosanquet-type empirical relation at the Darcy scale is found between the normalized effective gas diffusion coefficient and the average Knudsen number. The present work will promote the understanding and modeling of gas diffusion in nanoporous materials and also provide an efficient platform for the optimization design of nanoporous systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Andisheh-Tadbir, M., El Hannach, M., Kjeang, E., Bahrami, M.: An analytical relationship for calculating the effective diffusivity of micro-porous layers. Int. J. Hydrogen Energy 40, 10242–10250 (2015)

    Article  Google Scholar 

  • Becker, J., Wieser, C., Fell, S., Steiner, K.: A multi-scale approach to material modeling of fuel cell diffusion media. Int J Heat Mass Tran 54, 1360–1368 (2011)

    Article  Google Scholar 

  • Berson, A., Choi, H.-W., Pharoah, J.G.: Determination of the effective gas diffusivity of a porous composite medium from the three-dimensional reconstruction of its microstructure. Phys. Rev. E 83, 026310 (2011)

    Article  Google Scholar 

  • Beskok, A., Karniadakis, G.E.: Report: a model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Thermophys. Eng. 3, 43–77 (1999)

    Article  Google Scholar 

  • Bhattacharya, S., Gubbins, K.E.: Fast method for computing pore size distributions of model materials. Langmuir 22, 7726–7731 (2006)

    Article  Google Scholar 

  • Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena. Wiley, New York (2002)

    Google Scholar 

  • Blundell, S.J., Blundell, K.M.: Concepts in Thermal Physics. Oxford University Press, Oxford (2009)

    Book  Google Scholar 

  • Bruggeman, V.D.: Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen. Ann. Phys. 416, 636–664 (1935)

    Article  Google Scholar 

  • Chai, Z., Huang, C., Shi, B., Guo, Z.: A comparative study on the lattice Boltzmann models for predicting effective diffusivity of porous media. Int. J. Heat. Mass. Trans. 98, 687–696 (2016)

    Article  Google Scholar 

  • Chan, C., Zamel, N., Li, X.G., Shen, J.: Experimental measurement of effective diffusion coefficient of gas diffusion layer/microporous layer in PEM fuel cells. Electrochim. Acta 65, 13–21 (2012)

    Article  Google Scholar 

  • Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-uniform Gases. Cambridge University Press, Cambridge (1953)

    Google Scholar 

  • Chen, S., Doolen, G.D.: Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329–364 (1998)

    Article  Google Scholar 

  • Chen, L., Wu, G., Holby, E.F., Zelenay, P., Tao, W.-Q., Kang, Q.: Lattice Boltzmann pore-scale investigation of coupled physical-electrochemical processes in c/pt and non-precious metal cathode catalyst layers in proton exchange membrane fuel cells. Electrochim. Acta 158, 175–186 (2015)

    Article  Google Scholar 

  • Cousins, T.A., Ghanbarian, B., Daigle, H.: Three-dimensional lattice boltzmann simulations of single-phase permeability in random fractal porous media with rough pore-solid interface. Transp. Porous Med. 122, 527–546 (2018)

    Article  Google Scholar 

  • Delerue, J., Perrier, E., Yu, Z., Velde, B.: New algorithms in 3D image analysis and their application to the measurement of a spatialized pore size distribution in soils. Phys. Chem. Earth Part A. 24, 639–644 (1999)

    Article  Google Scholar 

  • d’Humieres, D., Ginzburg, I., Krafczyk, M., Lallemand, P., Luo, L.S.: Multiple-relaxation-time lattice Boltzmann models in three dimensions. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 360, 437–451 (2002)

    Article  Google Scholar 

  • Dreyer, J.A.H., Riefler, N., Pesch, G.R., Karamehmedović, M., Fritsching, U., Teoh, W.Y., Mädler, L.: Simulation of gas diffusion in highly porous nanostructures by direct simulation Monte Carlo. Chem. Eng. Sci. 105, 69–76 (2014)

    Article  Google Scholar 

  • El Hannach, M., Singh, R., Djilali, N., Kjeang, E.: Micro-porous layer stochastic reconstruction and transport parameter determination. J. Power Sources 282, 58–64 (2015)

    Article  Google Scholar 

  • Ellis, M.W., Von Spakovsky, M.R., Nelson, D.J.: Fuel cell systems: efficient, flexible energy conversion for the 21st century. Proc. IEEE 89, 1808–1818 (2001)

    Article  Google Scholar 

  • Fei, F., Fan, J., Jiang, J.: Solid wall effect on the transport coefficients of gases. Sci. China Phys. Mech. Astron. 55, 927–932 (2012)

    Article  Google Scholar 

  • Froning, D., Brinkmann, J., Reimer, U., Schmidt, V., Lehnert, W., Stolten, D.: 3D analysis, modeling and simulation of transport processes in compressed fibrous microstructures, using the Lattice Boltzmann method. Electrochim. Acta 110, 325–334 (2013)

    Article  Google Scholar 

  • Froning, D., Yu, J., Gaiselmann, G., Reimer, U., Manke, I., Schmidt, V., Lehnert, W.: Impact of compression on gas transport in non-woven gas diffusion layers of high temperature polymer electrolyte fuel cells. J. Power Sour. 318, 26–34 (2016)

    Article  Google Scholar 

  • Galinsky, M., Sénéchal, U., Breitkopf, C.: The impact of microstructure geometry on the mass transport in artificial pores: a numerical approach. Model. Simul. Eng. 2014, 1–7 (2014)

    Article  Google Scholar 

  • Haynes, W.M., Lide, D.R., Bruno, T.J.: CRC Handbook of Chemistry and Physics. CRC Press, New York (2012)

    Google Scholar 

  • He, X., Guo, Y., Li, M., Pan, N., Wang, M.: Effective gas diffusion coefficient in fibrous materials by mesoscopic modeling. Int. J. Heat. Mass. Trans. 107, 736–746 (2017)

    Article  Google Scholar 

  • Hussain, M., Tian, E., Cao, T.-F., Tao, W.-Q.: Pore-scale modeling of effective diffusion coefficient of building materials. Int. J. Heat. Mass. Trans. 90, 1266–1274 (2015)

    Article  Google Scholar 

  • Inoue, G., Kawase, M.: Effect of porous structure of catalyst layer on effective oxygen diffusion coefficient in polymer electrolyte fuel cell. J. Power Sources 327, 1–10 (2016a)

    Article  Google Scholar 

  • Inoue, G., Kawase, M.: Understanding formation mechanism of heterogeneous porous structure of catalyst layer in polymer electrolyte fuel cell. Int. J. Hydrogen Energy 41, 21352–21365 (2016b)

    Article  Google Scholar 

  • Inoue, G., Yokoyama, K., Ooyama, J., Terao, T., Tokunaga, T., Kubo, N., Kawase, M.: Theoretical examination of effective oxygen diffusion coefficient and electrical conductivity of polymer electrolyte fuel cell porous components. J. Power Sour. 327, 610–621 (2016)

    Article  Google Scholar 

  • Ismail, M.S., Ingham, D.B., Hughes, K.J., Ma, L., Pourkashanian, M.: Effective diffusivity of polymer electrolyte fuel cell gas diffusion layers: an overview and numerical study. Int. J. Hydrogen Energy 40, 10994–11010 (2015)

    Article  Google Scholar 

  • Joshi, A.S., Peracchio, A.A., Grew, K.N., Chiu, W.K.S.: Lattice Boltzmann method for multi-component, non-continuum mass diffusion. J. Phys. D Appl. Phys. 40, 7593–7600 (2007a)

    Article  Google Scholar 

  • Joshi, A.S., Peracchio, A.A., Grew, K.N., Chiu, W.K.: Lattice Boltzmann method for continuum, multi-component mass diffusion in complex 2D geometries. J. Phys. D Appl. Phys. 40, 2961 (2007b)

    Article  Google Scholar 

  • Kärger, J., Ruthven, D.M., Theodorou, D.N.: Diffusion in Nanoporous Materials. Wiley, Weinheim (2012)

    Book  Google Scholar 

  • Kaviany, M.: Principles of Heat Transfer in Porous Media. Springer, New York (1995)

    Book  Google Scholar 

  • Kim, S.H., Pitsch, H.: Reconstruction and effective transport properties of the catalyst layer in PEM fuel cells. J. Electrochem. Soc. 156, B673–B681 (2009)

    Article  Google Scholar 

  • Kim, S.H., Pitsch, H., Boyd, I.D.: Lattice Boltzmann modeling of multicomponent diffusion in narrow channels. Phys. Rev. E 79, 016702 (2009)

    Article  Google Scholar 

  • Krishna, R., van Baten, J.M.: Investigating the validity of the Bosanquet formula for estimation of diffusivities in mesopores. Chem. Eng. Sci. 69, 684–688 (2012)

    Article  Google Scholar 

  • Lange, K.J., Sui, P.-C., Djilali, N.: Pore scale simulation of transport and electrochemical reactions in reconstructed PEMFC catalyst layers. J. Electrochem. Soc. 157, B1434 (2010)

    Article  Google Scholar 

  • Lange, K.J., Sui, P.-C., Djilali, N.: Pore scale modeling of a proton exchange membrane fuel cell catalyst layer: effects of water vapor and temperature. J. Power Sour. 196, 3195–3203 (2011)

    Article  Google Scholar 

  • Lange, K.J., Sui, P.-C., Djilali, N.: Determination of effective transport properties in a PEMFC catalyst layer using different reconstruction algorithms. J. Power Sour. 208, 354–365 (2012)

    Article  Google Scholar 

  • Litster, S., Epting, W., Wargo, E., Kalidindi, S., Kumbur, E.: Morphological analyses of polymer electrolyte fuel cell electrodes with nano-scale computed tomography imaging. Fuel Cells 13, 935–945 (2013)

    Google Scholar 

  • Luo, L.-S., Girimaji, S.S.: Theory of the lattice Boltzmann method: two-fluid model for binary mixtures. Phys. Rev. E 67, 036302 (2003)

    Article  Google Scholar 

  • Ma, Q., Chen, Z.: Lattice Boltzmann simulation of multicomponent noncontinuum diffusion in fractal porous structures. Phys. Rev. E 92, 013025 (2015)

    Article  Google Scholar 

  • Nanjundappa, A., Alavijeh, A.S., El Hannach, M., Harvey, D., Kjeang, E.: A customized framework for 3-D morphological characterization of microporous layers. Electrochim. Acta 110, 349–357 (2013)

    Article  Google Scholar 

  • Pollard, W.G., Present, R.D.: On gaseous self-diffusion in long capillary tubes. Phys. Rev. 73, 762–774 (1948)

    Article  Google Scholar 

  • Shou, D., Fan, J., Mei, M., Ding, F.: An analytical model for gas diffusion though nanoscale and microscale fibrous media. Microfluid. Nanofluid. 16, 381–389 (2014)

    Article  Google Scholar 

  • Siddique, N., Liu, F.: Process based reconstruction and simulation of a three-dimensional fuel cell catalyst layer. Electrochim. Acta 55, 5357–5366 (2010)

    Article  Google Scholar 

  • Singh, R., Akhgar, A.R., Sui, P.C., Lange, K.J., Djilali, N.: Dual-Beam FIB/SEM characterization, statistical reconstruction, and pore scale modeling of a PEMFC catalyst layer. J. Electrochem. Soc. 161, F415–F424 (2014)

    Article  Google Scholar 

  • Tomadakis, M.M., Sotirchos, S.V.: Effective Kundsen diffusivities in structures of randomly overlapping fibers. AIChE J. 37, 74–86 (1991)

    Article  Google Scholar 

  • Tomadakis, M.M., Sotirchos, S.V.: Ordinary and transition regime diffusion in random fiber structures. AIChE J. 39, 397–412 (1993)

    Article  Google Scholar 

  • Wang, M.: Structure effects on electro-osmosis in microporous media. J. Heat. Trans. T ASME 134, 051020 (2012)

    Article  Google Scholar 

  • Wang, M., Pan, N.: Predictions of effective physical properties of complex multiphase materials. Mater. Sci. Eng. R Rep. 63, 1–30 (2008)

    Article  Google Scholar 

  • Wang, M., Wang, J., Pan, N., Chen, S.: Mesoscopic predictions of the effective thermal conductivity for microscale random porous media. Phys. Rev. E 75, 036702 (2007a)

    Article  Google Scholar 

  • Wang, M., He, J., Yu, J., Pan, N.: Lattice Boltzmann modeling of the effective thermal conductivity for fibrous materials. Int. J. Thermal Sci. 46, 848–855 (2007b)

    Article  Google Scholar 

  • Wang, Z., Guo, Y., Wang, M.: Permeability of high-Kn real gas flow in shale and production prediction by pore-scale modeling. J. Na. Gas Sci. Eng. 28, 328–337 (2016)

    Article  Google Scholar 

  • Yang, F., Gu, J., Ye, L., Zhang, Z., Rao, G., Liang, Y., Wen, K., Zhao, J., Goodenough, J.B., He, W.: Justifying the significance of Knudsen diffusion in solid oxide fuel cells. Energy 95, 242–246 (2016)

    Article  Google Scholar 

  • Yoshida, H., Nagaoka, M.: Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation. J. Comput. Phys. 229, 7774–7795 (2010)

    Article  Google Scholar 

  • Yu, Z., Carter, R.N.: Measurement of effective oxygen diffusivity in electrodes for proton exchange membrane fuel cells. J. Power Sour. 195, 1079–1084 (2010)

    Article  Google Scholar 

  • Yu, Z., Carter, R.N., Zhang, J.: Measurements of pore size distribution, porosity, effective oxygen diffusivity, and tortuosity of PEM fuel cell electrodes. Fuel Cells 12, 557–565 (2012)

    Article  Google Scholar 

  • Yuan, J.L., Sunden, B.: On mechanisms and models of multi-component gas diffusion in porous structures of fuel cell electrodes. Int J Heat Mass Tran 69, 358–374 (2014)

    Article  Google Scholar 

  • Zamel, N., Li, X.: Effective transport properties for polymer electrolyte membrane fuel cells—with a focus on the gas diffusion layer. Prog. Energy Combust. Sci. 39, 111–146 (2013)

    Article  Google Scholar 

  • Zamel, N., Becker, J., Wiegmann, A.: Estimating the thermal conductivity and diffusion coefficient of the microporous layer of polymer electrolyte membrane fuel cells. J. Power Sources 207, 70–80 (2012)

    Article  Google Scholar 

  • Zhang, X., Gao, Y., Ostadi, H., Jiang, K., Chen, R.: Modelling water intrusion and oxygen diffusion in a reconstructed microporous layer of PEM fuel cells. Int. J. Hydrogen Energy 39, 17222–17230 (2014)

    Article  Google Scholar 

Download references

Acknowledgements

The authors appreciate helpful discussions with Prof. N. Pan. This work is financially supported by the NSF grant of China (No. U1562217) and the National Science and Technology Major Project on Oil and Gas (No. 2017ZX05013001).

Author information

Author notes

    Authors and Affiliations

    1. Department of Engineering Mechanics and CNMM, Tsinghua University, Beijing, 100084, China

      Yangyu Guo, Xinting He, Wenzheng Huang & Moran Wang

    Authors
    1. Yangyu Guo
      View author publications

      You can also search for this author in PubMed Google Scholar

    2. Xinting He
      View author publications

      You can also search for this author in PubMed Google Scholar

    3. Wenzheng Huang
      View author publications

      You can also search for this author in PubMed Google Scholar

    4. Moran Wang
      View author publications

      You can also search for this author in PubMed Google Scholar

    Corresponding author

    Correspondence to Moran Wang.

    Additional information

    Yangyu Guo and Xinting He have contributed equally to this work.

    Rights and permissions

    Reprints and permissions

    About this article

    Check for updates. Verify currency and authenticity via CrossMark

    Cite this article

    Guo, Y., He, X., Huang, W. et al. Microstructure Effects on Effective Gas Diffusion Coefficient of Nanoporous Materials. Transp Porous Med 126, 431–453 (2019). https://doi.org/10.1007/s11242-018-1165-4

    Download citation

    • Received:

    • Accepted:

    • Published:

    • Issue Date:

    • DOI: https://doi.org/10.1007/s11242-018-1165-4

    Keywords

    Search

    Navigation